Clique Irreducibility of Some Iterative Classes of Graphs

In this paper, two notions, the clique irreducibility and clique vertex
irreducibility are discussed. A graph G is clique irreducible if every
clique in G of size at least two, has an edge which does not lie in any
other clique of G and it is clique vertex irreducible if every clique in G
has a vertex which does not lie in any other clique of G. It is proved
that L(G) is clique irreducible if and only if every triangle in G has a
vertex of degree two. The conditions for the iterations of line graph,
the Gallai graphs, the anti-Gallai graphs and its iterations to be clique
irreducible and clique vertex irreducible are also obtained.